Throughout this part, $n$ denotes an integer greater than or equal to 3, and $p = [(n+1)/2]$ is the integer part of $(n+1)/2$.
Let $\lambda_1, \lambda_2, \lambda_3$ be three real numbers satisfying $$\lambda_1 \geqslant \lambda_2 \geqslant \lambda_3 \quad \lambda_1 - \lambda_2 - 2\lambda_3 \geqslant 0 \quad 2\lambda_1 + \lambda_2 - \lambda_3 \geqslant 0 \tag{III.3}$$
We define the Hankel matrix $M = H(a,b,c,b,a) = \begin{pmatrix} a & b & c \\ b & c & b \\ c & b & a \end{pmatrix}$, where $a, b, c$ are real.
What can be deduced from the previous result, regarding condition III.3 in the case $n = 3$? Using an ordered triplet $(\lambda, 1, 1)$, show that for $n = 3$, condition III.1 is not sufficient.

Say, by briefly justifying the answer, whether the following assertions are correct for all $A , B \in M _ { n } ( \mathbb { C } ) , \mu \in \mathbb { C }$. i) $\rho ( \mu A ) = | \mu | \rho ( A )$. ii) $\rho ( A + B ) \leqslant \rho ( A ) + \rho ( B )$. iii) $\rho ( A B ) \leqslant \rho ( A ) \rho ( B )$. iv) For $P \in M _ { n } ( \mathbb { C } )$ invertible, $\rho \left( P ^ { - 1 } A P \right) = \rho ( A )$. v) $\rho \left( { } ^ { t } A \right) = \rho ( A )$.

In this subsection, $E$ is a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Let $u$ be an endomorphism of $E$ such that $u^2 = \operatorname{Id}_E$. Show that $u$ is a permutation endomorphism if and only if $\operatorname{Tr}(u)$ is a natural integer.

In this subsection, $E$ is a $\mathbb{C}$-vector space of dimension $n \geq 1$. We say that an endomorphism $u$ of $E$ is a permutation endomorphism if there exists a basis $(e_1, \ldots, e_n)$ of $E$ and a permutation $\sigma \in \mathfrak{S}_n$ such that $u(e_j) = e_{\sigma(j)}$ for all $j \in \llbracket 1, n \rrbracket$.
Study whether the equivalence of the previous question holds when we replace the hypothesis $u^2 = \operatorname{Id}_E$ by $u^k = \operatorname{Id}_E$ for $k = 3$, then for $k = 4$.

Suppose $n \geqslant 3$. Give an example of a matrix in $\mathcal { M } _ { n } ( \mathbb { R } )$ with zero trace and zero determinant, but not nilpotent.

Show that the following three assertions are equivalent
(i) $R_u = +\infty$,
(ii) $\mathbb{M}_n(u) = \mathscr{M}_n(\mathbb{C})$,
(iii) $\mathbb{M}_n(u) \neq \emptyset$ and $\forall A \in \mathbb{M}_n(u), \forall B \in \mathbb{M}_n(u), A + B \in \mathbb{M}_n(u)$, and give an example of a sequence $u$ satisfying these three assertions and such that $u_k \neq 0$ for every $k \in \mathbb{N}$.

Let $A$ be a square matrix of real numbers such that $A ^ { 4 } = A$. Which of the following is true for every such $A$?
(A) $\operatorname { det } ( A ) \neq - 1$
(B) $A$ must be invertible.
(C) $A$ can not be invertible.
(D) $A ^ { 2 } + A + I = 0$ where $I$ denotes the identity matrix.

For $3 \times 3$ matrices $M$ and $N$, which of the following statement(s) is (are) NOT correct?
(A) $\quad N ^ { T } M N$ is symmetric or skew symmetric, according as $M$ is symmetric or skew symmetric
(B) $M N - N M$ is skew symmetric for all symmetric matrices $M$ and $N$
(C) $M N$ is symmetric for all symmetric matrices $M$ and $N$
(D) $\quad ( \operatorname { adj } M ) ( \operatorname { adj } N ) = \operatorname { adj } ( M N )$ for all invertible matrices $M$ and $N$

Let $M = \left( a _ { i j } \right) , i , j \in \{ 1,2,3 \}$, be the $3 \times 3$ matrix such that $a _ { i j } = 1$ if $j + 1$ is divisible by $i$, otherwise $a _ { i j } = 0$. Then which of the following statements is(are) true?
(A) $M$ is invertible
(B) There exists a nonzero column matrix $\left( \begin{array} { l } a _ { 1 } \\ a _ { 2 } \\ a _ { 3 } \end{array} \right)$ such that $M \left( \begin{array} { l } a _ { 1 } \\ a _ { 2 } \\ a _ { 3 } \end{array} \right) = \left( \begin{array} { l } - a _ { 1 } \\ - a _ { 2 } \\ - a _ { 3 } \end{array} \right)$
(C) The set $\left\{ X \in \mathbb { R } ^ { 3 } : M X = \mathbf { 0 } \right\} \neq \{ \mathbf { 0 } \}$, where $\mathbf { 0 } = \left( \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right)$
(D) The matrix $( M - 2 I )$ is invertible, where $I$ is the $3 \times 3$ identity matrix

Let $A$ and $B$ be real matrices of the form $\left[\begin{array}{ll}\alpha & 0 \\ 0 & \beta\end{array}\right]$ and $\left[\begin{array}{ll}0 & \gamma \\ \delta & 0\end{array}\right]$, respectively. Statement 1: $AB - BA$ is always an invertible matrix. Statement 2: $AB - BA$ is never an identity matrix.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is false, Statement 2 is true.
(3) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.

Let $A$ be any $3 \times 3$ invertible matrix. Then which one of the following is not always true?
(1) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | ^ { 2 } \cdot ( \operatorname { adj } ( \mathrm {~A} ) ) ^ { - 1 }$
(2) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | \cdot ( \operatorname { adj } ( \mathrm { A } ) ) ^ { - 1 }$
(3) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | \cdot A$
(4) $\operatorname { adj } ( \mathrm { A } ) = | A | \cdot A ^ { - 1 }$

Let $A$ be a $2 \times 2$ real matrix with entries from $\{0, 1\}$ and $|A| \neq 0$. Consider the following two statements: $(P)$ If $A \neq I_{2}$, then $|A| = -1$ $(Q)$ If $|A| = 1$, then $\operatorname{tr}(A) = 2$ Where $I_{2}$ denotes $2 \times 2$ identity matrix and $\operatorname{tr}(A)$ denotes the sum of the diagonal entries of $A$. Then
(1) $(P)$ is false and $(Q)$ is true
(2) Both $(P)$ and $(Q)$ are false
(3) $(P)$ is true and $(Q)$ is false
(4) Both $(P)$ and $(Q)$ are true

Which of the following matrices can NOT be obtained from the matrix $\begin{pmatrix} -1 & 2 \\ 1 & -1 \end{pmatrix}$ by a single elementary row operation?
(1) $\begin{pmatrix} 0 & 1 \\ 1 & -1 \end{pmatrix}$
(2) $\begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$
(3) $\begin{pmatrix} -1 & 2 \\ -2 & 7 \end{pmatrix}$
(4) $\begin{pmatrix} -1 & 2 \\ -1 & 3 \end{pmatrix}$

Let $a , b , c , d , r , s , t$ all be real numbers. It is known that three non-zero vectors $\vec { u } = ( a , b , 0 )$, $\vec { v } = ( c , d , 0 )$, and $\vec { w } = ( r , s , t )$ in coordinate space satisfy the dot products $\vec { w } \cdot \vec { u } = \vec { w } \cdot \vec { v } = 0$. Consider the $3 \times 3$ matrix $A = \left[ \begin{array} { l l l } a & b & 0 \\ c & d & 0 \\ r & s & t \end{array} \right]$. Select the correct options.
(1) If $\vec { u } \cdot \vec { v } = 0$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(2) If $t \neq 0$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(3) If there exists a vector $\overrightarrow { w ^ { \prime } }$ satisfying $\overrightarrow { w ^ { \prime } } \cdot \vec { u } = \overrightarrow { w ^ { \prime } } \cdot \vec { v } = 0$ and cross product $\overrightarrow { w ^ { \prime } } \times \vec { w } \neq \overrightarrow { 0 }$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(4) If for any three real numbers $e , f , g$, the vector $( e , f , g )$ can be expressed as a linear combination of $\vec { u } , \vec { v } , \vec { w }$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(5) If the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$, then the determinant of $A$ is not equal to 0

On the coordinate plane, let $A$ and $B$ denote the rotation matrices for clockwise and counterclockwise rotation by $90^{\circ}$ about the origin respectively. Let $C$ and $D$ denote the reflection matrices with reflection axes $x = y$ and $x = -y$ respectively. Select the correct options.
(1) $A$ and $C$ map the point $(1,0)$ to the same point
(2) $A = -B$
(3) $C = D^{-1}$
(4) $AB = CD$
(5) $AC = BD$

Let the second-order matrices $A = \left[ \begin{array} { l l } 1 & 0 \\ 1 & 0 \end{array} \right], B = \left[ \begin{array} { l l } 0 & 1 \\ 0 & 1 \end{array} \right]$. Select the correct options.
(1) $A ^ { 2 } = A$
(2) $A + B = B + A$
(3) $A B = B A$
(4) $( A - B ) ^ { 2 } = A ^ { 2 } - 2 A B + B ^ { 2 }$
(5) $( A + B ) ^ { 2 } = 2 ( A + B )$